Solutions of Nonlinear Polyharmonic Equation with Periodic Potential

TL;DR

This paper investigates quasi-periodic solutions to a nonlinear polyharmonic equation with periodic potential, establishing the existence of a large non-resonant set where solutions resemble plane waves at high frequencies.

Contribution

It proves the existence of a broad non-resonant set of wave vectors for which solutions approximate plane waves in a nonlinear polyharmonic setting.

Findings

Existence of a large non-resonant set ${ mff G}$ in $ ^n$

Solutions asymptotically close to plane waves for large wave vectors

Extension of linear spectral theory to nonlinear polyharmonic equations

Abstract

Quasi-periodic solutions of a nonlinear periodic polyharmonic equation in $\R^n$, $n>1$, are studied. It is proven that there is an extensive "non-resonant" set ${\mathcal G}\subset \R^n$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $.